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A193856 Triangular array: the fission of (p(n,x)) by ((2x+1)^n), where p(n,x)=(x+1)^n.

%I A193856 #12 Dec 26 2023 09:54:57
%S A193856 1,1,5,1,8,19,1,11,43,65,1,14,76,194,211,1,17,118,422,793,665,1,20,
%T A193856 169,776,2059,3044,2059,1,23,229,1283,4387,9221,11191,6305,1,26,298,
%U A193856 1970,8236,22382,38854,39878,19171,1,29,376,2864,14146,47090,106000
%N A193856 Triangular array:  the fission of (p(n,x)) by ((2x+1)^n), where p(n,x)=(x+1)^n.
%C A193856 See A193842 for the definition of fission of two sequences of polynomials or triangular arrays.
%F A193856 From _Peter Bala_, Jul 16 2013: (Start)
%F A193856 T(n,k) = sum {i = 0..k} (-1)^k*binomial(n+1,k-i)*(-3)^(k-i) for 0 <= k <= n.
%F A193856 O.g.f.: 1/( (1 - 2*x*t)*(1 - (3*x + 1)*t) )= 1 + (1 + 5*x)*t + (1 + 8*x + 19*x^2)*t^2 + .... Cf. A193860.
%F A193856 The n-th row polynomial R(n,x) = 1/(x + 1)*( (3*x + 1)^(n+1) - (2*x)^(n+1) ). (End)
%F A193856 T(n, k) = 3^k*binomial(n+1, k)*hypergeom([1, -k], [n-k+2], 1/3). - _Peter Luschny_, Nov 19 2018
%e A193856 First six rows:
%e A193856 1
%e A193856 1...5
%e A193856 1...8....19
%e A193856 1...11...43....65
%e A193856 1...14...76....194...211
%e A193856 1...17...118...422...793...665
%p A193856 T := (n,k) -> 3^k*binomial(n+1,k)*hypergeom([1,-k],[n-k+2], 1/3):
%p A193856 for n from 0 to 6 do seq(simplify(T(n,k)), k=0..n) od; # _Peter Luschny_, Nov 19 2018
%t A193856 z = 10;
%t A193856 p[n_, x_] := (2 x + 1)^n;
%t A193856 q[n_, x_] := (x + 1)^n;
%t A193856 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A193856 p1[n_, 0] := p[n, x] /. x -> 0;
%t A193856 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A193856 h[n_] := CoefficientList[d[n, x], {x}]
%t A193856 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A193856 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A193856 *)
%t A193856 TableForm[Table[h[n], {n, 0, z}]]
%t A193856 Flatten[Table[h[n], {n, -1, z}]]   (* A193857 *)
%Y A193856 Cf. A193842, A193857, A193860.
%K A193856 nonn,tabl
%O A193856 0,3
%A A193856 _Clark Kimberling_, Aug 07 2011